> Well, 100% of the emitted radiation hits the ground in terms of total power, but the radar is looking for the _closest_ reflection, which is some relatively small solid angle straight down.

This fixed angle paints a larger and larger circle as altitude increases. If you double your altitude, you quadruple the size of the surface that's reflecting radar back at you.

A bunch of stuff cancels out and you end up with 1/r^2 at the receiver, which is what we started with.